Finite Group Databasebeta

← #117939

Subgroups — #117939

Order 512.

571 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20826C211yesmaximal, characteristic
2#17672C211yesmaximal, characteristic
3#17590C211yesmaximal, characteristic
4C2×#1493C211yesmaximal, characteristic
5#8121C211yesmaximal, characteristic
6#7962C211yesmaximal, characteristic
7#7898C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2283C2211yescharacteristic
15#1848—21no—
16#1837—21no—
17#1810—22no—
18C4×#108—22no—
19#1537—22no—
20#1531—22no—
21#1528—22no—
22#1518—22no—
23#1511—22no—
24#1506—22no—
25#1493—22no—
26#1493C414yescomplemented
27C2×Q8⋊C8—21no—
28C2×Q8⋊C8—21no—
29#1307—22no—
30#1286—22no—
31#1279—22no—
32C22×M16—21no—
33C23×C8—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533D412yes—
42C2×C4⋊Q8Q811yescharacteristic
43C2×C4⋊Q8Q811yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432D412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63C2×#108—24no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423C2×C412yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73#407—21no—
74#407D412yes—
75C2×C22⋊C8—21no—
76C4×M16—21no—
77C4×M16D412yes—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8C2×C412yes—
83C2×C4×C8D411yescharacteristic
84C2×C4×C8D411yescharacteristic
85C2×C4×C8D412yes—
86#383—21no—
87#382D412yes—
88#381—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C22⋊C4—21no—
92#376—21no—
93#341—24no—
94#330—28no—
95#328—28no—
96Q8⋊C8—24no—
97Q8⋊C8—24no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—41no—
107C2×M16—82no—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8—44no—
113C22×C8—41no—
114C22×C8—41no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8C4⋊C412yes—
122C4⋊Q8C4⋊C412yes—
123#125—22no—
124#125—22no—
125#125—41no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143#117—22no—
144#117—22no—
145#117—41no—
146#117—41no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×Q811yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170#108—48no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187#97—22no—
188#97—22no—
189#97—41no—
190#97—41no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—21no—
195C4×C8—21no—
196C4×C8—41no—
197C4×C8C22⋊C412yes—
198C4×C8C22⋊C412yes—
199#95—22no—
200#95—22no—
201#95—41no—
202#95—41no—
203C24#12411yescharacteristic
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—44no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—82no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4#11711yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4#12011yescharacteristic
221C22×C4#12011yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4C22⋊Q811yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12611yescharacteristic
235M16—44no—
236M16—44no—
237M16—82no—
238M16—82no—
239C2×C8—24no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—82no—
250C2×C8—82no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—82no—
268C42—42no—
269C42—42no—
270C42—82no—
271C42C2×C22⋊C411yescharacteristic
272C42C2×C4⋊C411yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42#12011yescharacteristic
276C42#12111yescharacteristic
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42C22⋊Q811yescharacteristic
281C42C22⋊Q811yescharacteristic
282C42C22⋊Q811yescharacteristic
283C42#12311yescharacteristic
284C42#12311yescharacteristic
285C42#12611yescharacteristic
286C42#12711yescharacteristic
287C23—22no—
288C23—22no—
289C23—41no—
290C23#38811yescharacteristic
291C23#39311yescharacteristic
292C23#39411yescharacteristic
293Q8—44no—
294Q8—82no—
295Q8—82no—
296Q8—82no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4#38011yescharacteristic
324C2×C4#38511yescharacteristic
325C2×C4#38511yescharacteristic
326C2×C4#38611yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#39011yescharacteristic
329C2×C4#39211yescharacteristic
330C2×C4#39411yescharacteristic
331C2×C4#39611yescharacteristic
332C2×C4#39611yescharacteristic
333C2×C4#39711yescharacteristic
334C2×C4#39911yescharacteristic
335C2×C4#47112yes—
336C2×C4#47312yes—
337C8—24no—
338C8—42no—
339C8—82no—
340C8—82no—
341C8—82no—
342C8—82no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—44no—
351C22#143911yescharacteristic
352C22#203112yes—
353C22#205212yes—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—82no—
364C4—82no—
365C4—162no—
366C4#181412yes—
367C4#197711yescharacteristic
368C4#197911yescharacteristic
369C4#206412yes—
370C2—22no—
371C2—22no—
372C2—41no—
373C2#1108111yescharacteristic
374C2#1110811yescharacteristic
375C2#1112811yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
26. #1493C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#208262
#176722
#175902
C2×#14932
#81212
#79622
#78982